machine design, Vol.4(2012) No.4, ISSN pp

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1 machine design, Vol.4(2012) No.4, ISSN pp SELF-EXCITED FLOW OSCILATIONS INSIDE A CAVITY Markéta PETŘÍKOVÁ 1, * - Pavel PEUKERT 1 - Pavel KRYŠTŮFEK 1 1 Technical University of Liberec, Faculty of Mechanical Engineering, Liberec, Czech republic Original scientific paper Received ( ); Revised ( ); Accepted ( ) Abstract: The paper shortly introduce the experimental setup of a hydrodynamic table and its possibilities on a case where is modeled a simplified wood-wind instrument and its self-excited oscillations. These results are further compared with performed numerical simulations in a commercial program environment. Key words: hydrodynamic analogy, flow, flow-simulations 1. INTRODUCTION Investigation of real flow processes in a gaseous environment is hard to perform with simple devices. Various methods with different grades of simplification were invented. One way is the use of a so called hydrodynamic table, where the investigated case is transformed to a quasi-two-dimensional flow. This analog method is relative simple and illustrative, with a high range of possible complex initial and boundary conditions. But the researcher had to respect always the fundamental rules, which are based on the phenomena similarities between the real and simplified case. The detailed information about the so called hydrodynamic analogy could be found in Merzkirch [1], Nožička [2] Goldstein [5] and White [6]. Fig.1. Photography of the hydrodynamic table. The computing performance which is available on actual computer systems also raises the significance of flow simulations. They are especially popular in the field of engineering applications, which demands a fast solution for the given case with minimal costs. There are commercial and open source solutions. The most popular are probably Ansys Fluent or Ansys CFX for the commercial sector and OpenFoam for the open source. New approaches were developed as example the company Nogrid GmbH, uses an approach based on point influence space instead of the classic mesh design. The main goal of this paper is the investigation of selfexcited flow oscillation of a flow stream inside a cavity in relation to the phenomena which appear in wood-wind instruments and leads to the sound production. It is an interesting demonstration case of the hydrodynamic table, but only a small set of possible application area. A second part of this work is the comparison of numerical and analog results to verify the computation approach. 2. MEASUREMENT PRINCIPLE The experiment was performed on a unique hydrodynamic table (Fig.1), which allows twodimensional flow visualization. For flow visualization is necessary to use an opaque and minimal reflecting fluid. For this purpose is used commonly black colored water by cellulose fibre color. This color is used for its intensive coloring ability and nature-friendly behavior. For correct flow streamline imaging is also necessary a high reflective medium which reliable create a uniform dust cover over the fluid surface. For this purpose is here used aluminium dust mixed with special surfactant fluids which prevent the aluminium particles from creating large indistributable clusters. The aluminium dust alone creates big and worse divisible cluster due the fine comminution technology, where the particle are covered with a near unnoticeable oil film. The principle explanation is easy to catch with the help of scheme depicted in Fig.2. The whole hydrodynamic table could be divided in three main parts. First comes water at a specified mass flow rate, to the inlet chamber where it is distributed equally to the working area where is placed the whole setup with the area of interest. At the inlet chamber is also mixed the coloring medium to the fluid. The prepared aluminium dust is then uniformly sprinkled, with a special tool, over the free liquid surface at the passage from the inlet chamber to the working area. At the working area should be installed the onset zone to the object of interest to get the proper inlet flow character. *Correspondence Author s Address: Technical University of Liberec, Faculty of Mechanical Engineering, Studentská 2, Liberec, Czech republic, marketa.petrikova@tul.cz

2 After it is the area of interest, where is placed the observed object. Here is necessary a correct illumination of the area to get the best performance. Over the tested object is placed a recording device to take the pictures. This could be realized via a photo or film camera, which depends on the actual needs. An important factor for good pictures is the proper adjustment of time, aperture and sensitivity of the camera, which is in according to the given flow velocity, illumination and the flow structures which should be resolved. The liquid flows after the pass of the working area to the outlet chamber. From here the liquid is leaded to the drain. The inlet an outlet chamber is equipped with small flood gates, which helps adjusting flow rates and liquid level height for the given part. 3. EXPERIMENTAL SETUP The setup for this experiment is simply arranged with the aspect to show the principle of woodwind instruments (Fig.3). In a simple manner could be said that air flows continually and stabile through the mouthpiece inside the instrument. From the mouthpiece flows the stream inside the instruments resonator. Inside the resonator the pressure rise until it exceed a threshold value and the air is forced out of the resonator. This is the basic periodic phenomena that take place inside the instrument and cause the audible sound. This phenomena is hardly to observe for a real case so it could be simplified for the use inside a hydrodynamic table. Fig.3. Woodwind instruments and an internal sketch Fig.2. Scheme of the experimental arrangement. On the hydrodynamic table could be performed also other visualization methods (more could be find in Merzkirch [1]) due to the high transparency glass bottom. According to the hydrodynamic analogy (Nožička [2], Golstein [5]) is there the possibility to obtain approximate values of pressure differences for given points. If the water level of two points is compared, the difference responds to the pressure difference over a computational relation. For these purposes was developed a special measuring device (Petříková [4]). It is based on the signalizing of the moment when a sharp measure tip reaches the water surface. This could be used for the mentioned water level measurements and also for recordings of periodic liquid surface level oscillations. The principle of the oscillation measurement comes from simple observation. With the oscillation of the stream commonly a depression appears and disappears on the surface. This effect is a direct result of the oscillation so it could be used by the special tool for collection of the contact cases over a specified time. The setup enable flow rates up to 500 liters per hour, which were lead to the working area which has a width of 1 m and length of 2 m and a liquid level of approximately 2 cm. That allows an acceptable range of geometrical variations that could be tested. It should be notify, that an area for the flow developing before and for observations behind the geometry should be reserved. With the hydrodynamic analogy could be the case transformed to a quasi-two-dimensional flow. It is necessary to remember the simplification and possible nuances from the real three-dimensional phenomena. Also a slowdown of the time scale is one of the benefits of this method. The experimental layout is given in Fig.4. For this experiment was more in the foreground the illustrative nature than a special case of a woodwind instrument and that is reflected in the sizes. inlet chamber 1000 mm 500 narrowing configurations (m : n) y K inlet channel working area 950 constant width part mm nozzle Fig.4. Scheme of the experimental arrangement and the hydrodynamic table The experiment was done for volumetric flow rates from 100 to 500 liters per minute with a step of 100. At the inlet chamber was the water colored to black and covered with the aluminium dispersion. The water flows then through a relative long channel to a nozzle. The long channel should ensure a developed uniform flow. The y T m gap (m) (2:1) (1:1) (1:2) (1:3) n cavity (n) outlet chamber y N movable crossbar (cavity bottom) 214

3 nozzle represents the mouthpiece, where the stream is accelerated and more concentrated. The main area of interest is the cavity with a defined offset from the nozzle outlet. That corresponds to the inner of the resonator body and the notch in the instrument. From the gap flows the liquid to a huge area which represents the surrounding atmosphere. Also that part was done with care to ensure the similar interactions like in the atmosphere and preventing significant reflection waves. The important geometrical dimensions for the experimental data are depicted in Fig.5. The size of the inlet channel is constant till the nozzle y K = 60 mm. The narrowest part of the nozzle is for all experiments y T = 20 mm. And the cavity width is every time the same y N = 60 mm. The gap size m and the cavity deepness n change during the experiments, to show their influence on the flow character. These changing parameters correspond to different wood-wind instrument tuning, because with the length changes the sound wave length of the instrument. The gap size m was chosen 70 and 140 mm. The deepness was chosen for specific ratios of the gap to the deepness size (m:n), because it reflex better the similarity of the phenomena, then some good looking values with hard empathize to the reality. This ratio is 0.5, 1, 2 and 3. V Inlet channel y K nozzle Fig.5. Detailed sketch of the significant geometrical dimensions of the experimental substitution of a woodwind instrument. m y T n y N Cavity - resonator cavity bottom l/hour. The frequency jump shows a clear state where rises the vortex generation and could be a special state of oscillations in a natural frequency for the given case. Fig.6. Dependence between the oscillation frequency and the characteristic dimension m + n [cm] for the gap m = 140 [mm] and different volumetric flow rates [l/h]. The next chart (Fig.7) shows thee dependence of the Strouhal number on the characteristic dimension for different flow rates. The Strouhal number is defined as L f Sh =, (1) w where L is the given characteristic dimension, f is the frequency of vortex shading, here equal with the oscillation frequency and w is the fluid velocity out coming from the nozzle. The dimensionless number could give us a good comparison for the individual cases. 4. RESULTS 4.1. Experimental results There were a lot of results from the experiments, for the gentle overview was chosen a representative case of data, where the gap size was m = 14 mm and the deepness varies from n = 0 to 42 mm. The visualisation part is in the next subchapter, because to made an approximate comparison. The following part shows results computed from the data collection made by the special tool for the liquid level measurements (Petrikova [4]). In the first chart (Fig.6) is the dependence of the stream oscillation frequency for the base volumetric flow rates and a different characteristic dimension. The characteristic dimension here is the sum of the size of the gap m and cavity deepness n. It is interesting to see that for ratios (m : n) up to 1:1,5 and behind 1:2.3 increases slightly the frequency with the flow rate (only for the size of 18 cm is to see an exception). Between this ratios could be observed a frequency small frequency jump, which starts with higher flow rates at a higher characteristic dimension, the only exception is the flow rate of 200 Fig.7. Dependence of Strouhals number Sh [-] on the characteristic dimension m + n [cm] for the gap m = 140 [mm] and different volumetric flow rates [l/h]. 215

4 The last chart (Fig.8) shows the dependence of the Strouhal number for the given Reynolds number for different cavity deepness. The Reynolds number is defined as L w Re =, (2) 216 ν where L is the same characteristic size, w the fluid velocity at the inlet and ν the kinematic viscosity. Only few deepness sizes were chosen for this chat to ensure the clear arrangement. From the chart could be estimated that the character of the flow changes proportionally to the characteristic size so that the point are change with the flow rate, but had for the same geometry a similar development. Only few anomalies could be observed for the deepness of 40 mm and 28 mm. This chart is the one that give the alignment between the air and liquid experimental process. Fig.8. Dependence of Strouhal's number on Reynolds number for the gap m = 140 [mm] and different cavity deepness n [cm] Experimental visualization and numerical comparison The numerical simulations were done by the software ANSYS Fluent 13. For the simulation was used a two-dimensional hybrid mesh, created in Gambit (Fig.10) and contain a sum of tri and tetragonal elements. As boundary conditions was used a constant velocity inlet at the nozzle inlet and a constant counter-pressure outlet at the area of the outlet chamber. The other geometrical boundary structures are defined as wall. It is necessary to mention that the velocity inlet have to be converted to the twodimensional form trough the dimensionless quantity for this case. The simulations were done by the SST k-ω incompressible model. Because the observed process is variable in time, the solver was set as unsteady with a time step of s, else were the oscillation of the stream not possible to observe. The convergence condition was selected for all monitored values Because the results represent a huge database only few results were chosen to give a comparison to the experimental visualization. For this purpose was chosen to show the vector flow field and the background color represents the absolute magnitude of the velocity. The comparable picture to the simulation is located under the one of the simulation. Fig.9. Computation mesh for the simulation made in Gambit The experimental visualization photographs were chosen to present in pictures under the concrete simulation, where is the flow similarity to see. The following three photographs represent the cycle of one oscillation of the stream. All three sets (Fig.11, Fig.13, Fig.15) were for the same geometry of the gap sizes of m = 140 mm and cavity deepness n = 140 mm. The flow rates vary in the following order 300, 400 and 500 liter per hour. Fig.10 and the Fig.11 set show the output for the flow rate of 300 l/h. There is a very good similarity between the simulation and the reality. On the picture set Fig.11 could be good observed the oscillation of the stream cycle. The first picture shows the emptying of the cavity. It influences the main stream which is turned aside by the overfilled liquid inside the cavity. In the next picture decreased the amount of the liquid inside the cavity so much, that the main stream is soaking inside the cavity and the refilling starts. In the next picture is shown the filing process, the amount of the liquid rises in the cavity. After a limit amount of the liquid is reached the emptying of the cavity starts again. The stream is deflected and the liquid flows out and the whole process starts again. Also here is good understandable the hydrodynamic analogy. As mentioned before, the liquid level difference corresponds to a specific pressure difference. There is a correlation between the potential energy and the pressure energy. So it could be conclude from the process which takes part in the hydrodynamic table to the real process. Like in the filling time rises the incompressible liquid amount and with it the height of the water level. Projected to the reality of a compressible flow, the mass amount in the cavity/resonator increase also and because there is a constant volume the pressure have to increase. After the reach of a limit value of the liquid level/pressure value, which is direct connected to the environment liquid level/pressure and main stream properties the cavity/resonator starts emptying. In the pictures is the effect hard to see, because it is only two-dimensional, but also on the liquid surface appearing waves, that in reality correspond to the sound waves.

5 Markéta Petříková, Pavel Peukert, Pavel Kryštůfek: Self-excited Oscilations Inside a Cavity; Fig.10. Numerical simulation for the gap 140 mm, ratio 1 and flow rate of 300 l/min. Fig.12. Numerical simulation for the gap 140 mm, ratio 1 and flow rate of 400 l/min. Fig.11. Experimental visualization set for the gap 140 mm, ratio 1 and flow rate of 300 l/min. Fig.13. Experimental visualization set for the gap 140 mm, ratio 1 and flow rate of 400 l/min. 217

6 5. CONCLUSION Fig.14. Numerical simulation for the gap 140 mm, ratio 1 and flow rate of 500 l/min. Only a small summary was presented from a huge measurement database to show the possibilities of the mentioned experimental approach. The chart in Fig.8 represents the analogy to the real air case. It should represent the dependence of the tone on the resonator deepness. That mean a minimal influence of the velocity on the oscillation, that fact is to see for some results, but other disagree. Also the Strouhals number should approximate double with the double of the characteristic size and could be described via a function. That fact could also be used in engineering for designing of devices with self-dosage without any other impulses. The experiment had also confirmed that for generating a periodical progress of the flow is sufficient a constant uniform flow without periodical forcing. The numerical analysis from Fig.10, 12 and 13 shows good agreements. Similarities could be find, especially for the main vortex inside the resonator and also the declination of the main stream. Only in Fig.13 is the inner vortex at the base of the cavity larger in the real case than in the experiment. That means that some improvements could be done in the future. It is planned for the future to increase the element amount for a more detailed computation and switch to a three-dimensional simulation with a phase boundary. That adaptation should agrees better to the real case. ACKNOWLEDGEMENT The authors gratefully acknowledge the support by the grand project SGS REFERENCES 218 Fig.15. Experimental visualization set for the gap 140 mm, ratio 1 and flow rate of 500 l/min [1] Merzkirch W. (1974). Flow visualization, Academic press, ISBN , New York and London. [2] Nožička J. (1967). Analogové metody v proudění, Academia Praha, Praha. [3] Šidlof P. (1972). Použití hydrodynamické analogie pro výzkum výměny obsahu válců spalovacích motorů, Vysoká škola strojní a textilní v Liberci, Thesis, Liberec. [4] Petříková M., Kneř J., Jerje P. (2010), Snímač hladiny elektricky vodivé kapaliny, zejména výšky hladiny a frekvence jejího kmitání. Úřad průmyslovho vlastnictví, č zápisu [5] Goldstein I., Richard J. (1983), Fluid mechanics measurements, Taylor and Francis, ISBN X, Philadelphia. [6] White F. M. (2001), Fluid Mechanics Fourth Edition, McGraw-Hill Education, ISBN [7] Ansys Inc. (2011), Documentation for ANSYS Fluent 13, ANSYS Inc., Canonsburg.